#366710

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(5,6)(7,8)(9,10,15,20,16,22,14,21)(11,23,25,17,13,32,26,19)(12,27,29,18,31,35,30,36)(24,33,39,37,28,34,38,40);(1,2,3,4)(5,6)(7,8)(9,11)(10,17)(12,24)(13,16)(14,26)(15,25)(18,33)(19,22)(20,32)(21,23)(27,40)(28,31)(29,39)(30,38)(34,36)(35,37);(2,4)(6,8)(9,12)(10,18)(11,24)(13,28)(14,29)(15,30)(16,31)(17,33)(19,34)(20,27)(21,35)(22,36)(23,37)(25,38)(26,39)(32,40);(9,13,16,11)(10,19,22,17)(12,28,31,24)(14,25,15,26)(18,34,36,33)(20,23,21,32)(27,37,35,40)(29,38,30,39)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273906);